
\section{Active Force Models}
\label{sec:forces}

We consider two models for the active forces: the first being the generic case where arbitrary forces can be specified and the second model which accounts for the contractility of the myocytes. For the generic case the force term is given by,

\begin{equation}
\label{e:genericForce} 
F(t) = M\mathbf{f}(t),
\end{equation}  

where, $\grbf{f}(t)$ are the nodal forces at time $t$.

To model the contractility of the myocytes given the fiber contractility $\grbf{s}$ as a function of space and time and the myocyte orientation $n$, we define the active stretch tensor $U=1+ \grbf{s}\, n\otimes n$, whose divergence results in a distributed active force of the form $\Div(\grbf{s}\, n\otimes n)$. The force term in this case is given by, 

\begin{equation}
\label{e:fiberForce} F(t) = A\grbf{s}(t), \quad A_{ij}= \int (n \otimes n) \Grad \phi_i \phi_j.
\end{equation} 
 
To reduce the computational cost for the calculations, we used a reduced-order model for $\grbf{s}$ and $\grbf{f}$ as a combination of B-splines bases in time and radial basis functions in space (Gaussians). The forces can then be written in terms of the B-spline basis, $B$ and the radial basis, $G$, and the control parameters $\grbf{\mu}$,

\begin{eqnarray}
	\grbf{f}(\grbf{x}, t) &=& \sum_{k=1}^3 \grbf{e}_k \sum_i G_i^k(\grbf{x}) \sum_j \grbf{\mu}_{ij} B_{j}(t), \\
	\grbf{s}(\grbf{x}, t) &=& \sum_i G_i(\grbf{x}) \sum_j \grbf{\mu}_{ij} B_{j}(t).
\end{eqnarray}

In the matrix form this defines the parametrization matrix $C$ is given by,

\begin{equation}
\label{e:C}
C_{xt, ij} = G_i(x) B_{j}(t).
\end{equation}

We can write the active forces in terms of the parametrization matrix $C$, as

\begin{eqnarray*}
F &=& M \grbf{f} = M C \grbf{\mu}, \\
F &=& A \grbf{s} = A C \grbf{\mu}.
\end{eqnarray*}

